THE ASTROPHYSICAL JOURNAL, 500 : 1009È1022, 1998 June 20
( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SPECTRAL LINES FOR POLARIZATION MEASUREMENTS OF THE CORONAL MAGNETIC FIELD. I.
THEORETICAL INTENSITIES
P. G. JUDGE
High Altitude Observatory, National Center for Atmospheric Research,1 P.O. Box 3000, Boulder, CO 80307-3000
Received 1997 November 10 ; accepted 1998 April 10
ABSTRACT
Infrared emission lines are potentially sensitive probes of components of the coronal vector magnetic
Ðeld, through the Zeeman e†ect, and on its direction projected onto the plane of the sky, through Ñuorescent polarization of scattered photospheric light. Prompted by the advent of sensitive infrared array
detectors, existing atomic data were reexamined to compile a complete list of coronal lines that may
yield a detectable Zeeman e†ect, through careful di†erential measurements of Stokes proÐles, at typical
coronal Ðeld strengths of order 10 G. ““ Average ÏÏ intensities were computed for a subset of promising
forbidden coronal lines. A representative coronal density structure was used. The distribution of plasma
with temperature was, at all heights in the corona, assumed to be that described by a standard di†erential emission measure from extreme-ultraviolet observations of the solar disk. E†ects of excitation by
photospheric radiation were included, as well as cascades from collisionally excited higher levels having
the same principal quantum number as the ground levels. The largest source of error in the computed
intensities lies in the form assumed for the emission measure distribution. The assumed density and temperature structure is too simple for detailed comparisons with observations of a particular coronal structure. Nevertheless, existing observed intensities are consistent with the calculations, which suggests that
the theoretical intensities of (as yet) unobserved lines can be used as a basis for further study. The
strongest predicted lines arise from magnetic dipole transitions within the ground terms of the 2sm2pn
and 3sm3pn, m \ 1, 2, n \ 1, . . . , 5, conÐgurations. The most promising lines lie between 1 and 10 km, the
lower limit being set by the need to detect small Ðeld strengths. The upper limit is set by the small
Einstein A-coefficients and the smaller intensities of the exciting photospheric light, both of which lead to
smaller forbidden line intensities. The most promising lines include [Fe XIII] 1.0747, 1.0798 km ; [Si X]
1.43 km ; [Si IX] 2.58, 3.93 km ; [Mg VIII] 3.03 km ; and [Mg VII] 5.50, 9.03 km. An aircraft experiment is
being prepared to obtain targeted portions of the coronal spectrum between 1 and 10 km during the
1998 February 26 eclipse, with the goal of detecting some of these promising lines. This work will help
toward the planning and development of efficient magnetographs, perhaps space-borne, for the routine
measurement of coronal magnetic Ðelds in the quiet and active Sun.
Subject headings : polarization È Sun : atmosphere È Sun : corona È Sun : infrared È
Sun : magnetic Ðelds
1.
INTRODUCTION
the solar wind outÑow. Such measurements contain little, if
any, useful information about conditions in the inner
corona, beyond the statement that the magnitude of the
Ðeld at 1 AU implies average Ðeld strengths close to 10 G at
the coronal base.
Remotely sensed determinations of coronal magnetic
Ðelds have been limited to special regions of the solar atmosphere and/or at special times. Radio observations of
unusually strong Ðeld regions (greater than 100 G) overlying active regions can yield information on projections of
the vector magnetic Ðeld B through the gyroresonance
mechanism. A substantial literature has evolved on this
subject since gyroresonance emission was predicted by
Ginzburg & Zheleznyakov (1961) (and in earlier papers in
Russian). The mechanism was conÐrmed through various
studies, based on comparisons between extreme-ultraviolet
(EUV) observations with Skylab and radio aperture synthesis work (e.g., Kundu, Schmahl, & Gerassimenko 1980 ;
Pallavicini, Sakuri, & Vaiana 1981 ; Webb et al. 1983 ;
Shibasaki et al. 1983), on the appearance of ““ ring ÏÏ structures and characteristic polarization signals in radio data
(Alissandrakis & Kundu 1982 ; Lang & Willson 1982) predicted in the presence of gyroresonance opacity by
Gelfreikh & Lubyshev (1979), and on the detection of individual ““ lines ÏÏ in the gyroresonant spectrum (Willson 1985).
The evidence for magnetic control of the SunÏs corona is
overwhelming, from early eclipse and coronagraphic
observations showing loop structures to present-day images
from the Y ohkoh satellite and SOHO spacecraft. It is abundantly clear, from observations of the 106 K coronal
plasma, that the magnetic Ðelds emerging from beneath the
SunÏs photosphere control essentially all aspects of coronal
structure and dynamics, from plasma-heating mechanisms
through to coronal evolution on the largest spatial scales
(for reviews, see Bray et al. 1991 ; Low 1994 ; Golub & Pasacho† 1997).
Unfortunately, it is equally clear that our empirical
knowledge of the controlling coronal magnetic Ðelds is
extremely limited. Direct determinations of coronal magnetic Ðelds in the ““ ordinary ÏÏ corona (i.e., the 106 K plasma
in the quiet Sun or surrounding active regions) are limited
to regions 100 R or more above the photosphere where
_
spacecraft can operate
(e.g., Erdos, Balogh, & Kota 1997 ;
Forsyth et al. 1997). In such regions, the plasma b is of order
1 or more, i.e., the energy density is dominated by plasma
motions and the Ðeld lines have already been ““ opened ÏÏ by
1 The National Center for Atmospheric Research is sponsored by the
National Science Foundation.
1009
1010
JUDGE
Recent papers have developed this area to the extent that
““ coronal magnetography ÏÏ of active regions using this
method is entering a phase of maturity (e.g., White, Kundu,
& Gopalswamy 1991 ; Gary & Hurford 1994 ; Schmelz et al.
1994 ; Brosius et al. 1997), with the higher spatial resolution
and spectral coverage possible today.
As for active regions, prominences must also be considered a special case, in which speciÐc thermal conditions
exist that permit application of the Hanle e†ect and the
Zeeman e†ect in ions of unusually low ionization degree
(see KoutchmyÏs 1995 review, for example). Magnetic Ðeld
determinations can also be made by Faraday rotation measurements of cosmic radio sources (e.g., Gelfreikh 1994). In
all of these cases, the regions of the corona that can be
measured are ““ special,ÏÏ being speciÐc locations determined
by special thermodynamic conditions, by special magnetic
conÐgurations, or by particular lines of sight through the
corona.
In the 106 K, low-b coronal plasma typically found in the
quiet Sun, there are only very weak signatures of the presence of the magnetic Ðeld in the emitted spectra. For
instance, thermal bremsstrahlung can be used to determine
longitudinal magnetic Ðeld strengths through observations
of left and right circularly polarized radio waves. These
observations are difficult owing to the extremely high sensitivity needed, and results are quite scarce (e.g., Gelfreikh
1994).
Currently, our view of the vector Ðeld in the 106 K corona
of the quiet Sun in fact comes almost entirely from extrapolation of photospheric magnetic Ðeld measurements. While
signiÐcant e†orts have been made to determine the morphology of the inferred coronal magnetic Ðelds through
comparisons of calculations with active region EUV plasma
loops in active regions (e.g., Schmieder et al. 1996), such
extrapolations are inherently limited by theoretical uncertainty in the models used owing to the ill-posed nature of
the problem (e.g., Low & Lou 1990). In the quiet Sun, the
problem is considerably worse owing to the weaker, less
ordered Ðelds there. Experimental determinations of the
magnetic Ðeld, B, in the 106 K corona are needed to put
coronal physics on a Ðrm empirical foundation. Such measurements are needed to address many crucial questions :
How reliable are extrapolations that are commonly used
today ? Do the Ðelds carry large-scale currents, and how are
these currents formed (e.g., Leka et al. 1996) ? How complex
or ““ braided ÏÏ is the Ðeld within a given loop, and how does
this inÑuence models of coronal heating (e.g., Parker 1988) ?
What kind of Ðeld conÐgurations lead to prominence formation (Low & Hundhausen 1995) ? What causes the
dynamic behavior seen in Ñares and coronal mass ejections
(e.g., Low 1994) ?
Two diagnostics of the 106 K coronal plasmaÏs magnetic
Ðeld exist that have not yet been proven useful : the Hanle
e†ect, for which the Lya line of hydrogen holds the most
promise (e.g., Fineschi et al. 1993), and the Zeeman e†ect.
The Ðrst method su†ers from the disadvantage that at
present there is no complete quantum theory to attack the
general problem of the frequency-dependent resonance
scattering process including quantum coherences between
sublevels. Nevertheless, a recent heuristic extension of the
standard theory by Landi DeglÏInnocenti et al. (1996) is
promising (Landi DeglÏInnocenti 1998).
The purpose of the present paper is to identify, through
theoretical calculations, a set of infrared forbidden coronal
Vol. 500
emission lines that can be expected to reveal properties of
the 106 K coronal plasmaÏs magnetic Ðeld of the quiet Sun
using Stokes polarimetry, through the Zeeman e†ect. The
line proÐles, which are observable not on the disk but above
the solar limb, in principle contain information on the
vector magnetic Ðeld through small but systematic line
shifts between the di†erent polarization states arising from
the Zeeman e†ect, which can be determined through careful
di†erential measurement techniques (Kuhn 1995). This
paper is devoted only to the calculation of intensities, comparisons with observations, identiÐcation of excitation
mechanisms, and a Ðrst assessment of the best emission
lines that promise to be useful. Later work will focus upon
simulation of the full Stokes vectors, including the e†ects of
the EarthÏs atmosphere and instrumental properties, in
order to identify the best lines, the kind of instrumentation
needed for future measurement e†orts, and problems concerning inversion of the polarization data to derive magnetic Ðelds. It is hoped that such lines may eventually yield
magnetic Ðeld determinations that are complementary to
the more established radio techniques, yielding information
on larger volumes of the corona and in weaker Ðeld regions.
This paper is organized as follows. First, a brief review is
given of previous attempts to measure coronal Ðeld
strengths, of the techniques that are available, and of the
potential advantages o†ered by selecting infrared transitions. Next, since previous work lacks completeness in
terms of atomic data and treatment of excitation processes,
detailed calculations are presented for speciÐc ions that are
likely to be of interest, based upon a complete survey of
interesting atomic transitions discussed in the Appendix.
Finally, comparisons with observations are made, and a list
of potentially useful lines is compiled.
2.
EMISSION LINES AS DIAGNOSTICS OF CORONAL
MAGNETIC FIELDS
Extrapolations of Ðelds measured at 1 AU indicate Ðeld
strengths of order 10 G in the quiet SunÏs corona. Such
estimates immediately reveal the well-known reason that
direct measurements of 106 K coronal plasma Ðelds have so
far eluded observers. Fields of this magnitude are notoriously difficult to measure from properties of radiation
emitted in 106 K plasmas (e.g., Harvey 1969). To see why,
we note that of those techniques that can be presently
applied to 106 K plasmas, there are just three that are
potentially useful :
1. The Hanle e†ect can be used in permitted lines, especially H Lya.
2. The Zeeman e†ect can in principle be used to determine line-of-sight projections of the vector magnetic Ðeld,
using di†erential measurements in circularly polarized light.
3. Linear polarization degree measurements for certain
forbidden coronal lines that are excited by the anisotropic
photospheric radiation can constrain the direction of the
magnetic Ðeld in the plane of the sky (POS). This will be
referred to henceforth as the ““ Ñuorescent polarization ÏÏ
method.
The Hanle e†ect will not be discussed further. The
Zeeman e†ect produces Stokes proÐles that are in the (very)
weak Ðeld limit. Doppler widths exceed the wavelength
shifts seen in di†erent Stokes parameters by several orders
of magnitude. The Doppler width of [Fe XIII] 10747 AŽ , for
example, is 3 orders of magnitude larger than the wave-
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
length shifts induced in di†erent states of circular polarization by a 10 G Ðeld component along the line of sight
(LOS). Thus, careful measurements di†erential in wavelength as well as polarization state are required (see Kuhn
1995) with high signal-to-noise ratios. Using circular polarization measurements, Harvey (1969) derived upper limits
of several tens of gauss for the LOS component ( o B o ) from
A
observations of the green line [Fe XIV] 5304 AŽ . Similarly, an
upper limit of 40 G was determined by Kuhn (1995) using
[Fe XIII] 10747 AŽ line proÐles observed with a 0.4 m coronagraph outside of eclipse.
The Ñuorescent polarization method has been successfully applied to observations of the [Fe XIII] 10747 AŽ
line (e.g., Arnaud & Newkirk 1987 and references therein).
The method relies on the fact that certain radiatively excited
transitions originate in resonant transitions between radiatively ““ aligned ÏÏ Zeeman substates (e.g., Sahal-Brechot
1977). It does not require wavelength resolution across line
proÐles, and it has intrinsically high fractional polarizations
(several tens of percent are possible ; e.g., see Sahal-Brechot
1977). Thus, these measurements are easier to make than
those needed for the Zeeman e†ect. A necessary (but not
sufficient) condition for the Ñuorescent polarization method
to work is that the line is excited, at least in part, by the
absorption of the anisotropic photospheric radiation. If the
excitation mechanism for a given line satisÐes a sufficiency
condition (see Sahal-Brechot 1977 for the particular case of
[Fe XIII]), then both methods can be applied simultaneously
to coronal forbidden lines.
In summary, although measurements have been made
that have constrained the direction of coronal magnetic
Ðelds in the POS, to date no magnetic Ðeld strength measurements have been made in the quiet SunÏs corona, owing
to difficulties in measuring small di†erential properties of
Stokes proÐles. The work of Kuhn (1995) is an important
Ðrst step along the path for developing coronal magnetometry beyond its modest beginning in the 1960s and 1970s.
His upper limit of 40 G, which was obtained with a coronagraphic instrument using a good, but not state-of-the-art,
detector, is close to expected Ðeld strengths. He emphasized
that the infrared (IR) spectral region can o†er signiÐcant
advantages over optical wavelengths. In particular, the
sensitivity of Doppler-broadened Stokes proÐles to the
Zeeman e†ect increases linearly with the wavelength of
the line. Recent improvements in the sensitivity and size of
IR detector arrays also open up the possibility for very
sensitive measurements over large projected volumes of the
corona.
3.
CALCULATIONS OF THE INFRARED LINE SPECTRUM OF
THE CORONA
3.1. T he Need for T heoretical Spectral Calculations
Our knowledge of the IR spectrum of the SunÏs corona is
surprisingly sparse. Only seven or so lines have been identiÐed in the IR region between 1 and 3 km (Olsen, Anderson,
& Stewart 1971 ; Kastner 1993 ; Kuhn 1995). This stands in
contrast to the large number of lines expected in this wavelength range and at longer wavelengths from compilations
of atomic energy level data of highly ionized ions (Edlen
1943 ; Rohrlich & Pecker 1963 ; Pryce 1964 ; Collins 1964 ;
Kaufman & Sugar 1986). This is due to the relative weakness of the coronal forbidden lines compared with the solar
disk brightness, to the difficulties in measuring IR lines
1011
under high, varying spectral background conditions, and to
the restricted windows in the atmospheric transmission of
IR radiation.
This situation motivates the use of theoretical calculations to explore and identify strong IR forbidden coronal
lines. However, calculations needed to compute a synthetic
coronal spectrum are generally unavailable. With interest in
data from the recently launched Infrared Space Observatory
satellite, Greenhouse et al. (1993) computed the properties
of a set of ions under conditions believed to be present in
the line-emitting regions of Seyfert galaxies. While aspects
of these calculations will be used below, their usefulness is
limited by the omission of two processes known to be
important in the formation of some forbidden lines in the
corona : photoexcitation by the anisotropic solar photospheric Ðeld and cascades from higher atomic levels excited
by electron collisions (e.g., Flower & Pineau des Foreüts
1973 ; Sahal-Brechot 1974). Both are crucial processes to
model correctly because not only are intensities greatly
inÑuenced, but the Ñuorescent polarization method relies
primarily on photoexcitation as a signiÐcant excitation
process. Calculations are therefore needed (1) to determine
which types of lines in which kinds of elements and ions are
expected to be strongest and (2) to perform detailed calculations for speciÐc cases, including all processes known to
be important, to produce synthetic forbidden line spectra of
the corona.
3.2. First Calculations
Before embarking on detailed calculations, it is appropriate (and necessary) to review and reÐne a large list of potentially interesting transitions (Kaufman & Sugar 1986). In
the Appendix, a ““ Ðrst pass ÏÏ through atomic data is
described. The survivors of this Ðrst cut, based on abundances listed in Table 1 and drawing heavily on the work of
Greenhouse et al. (1993), are listed in Table 2. These are
potentially bright (mostly IR) lines arising from transitions
within the 2sm2pn and 3sm3pn, m \ 1, 2, n \ 1, . . . , 5 conÐgurations in abundant ions. All other transitions are likely to
be substantially weaker on the basis of small branching
TABLE 1
ADOPTED ABUNDANCES
Element
Ne . . . . . .
Na . . . . . .
Mg . . . . . .
Al . . . . . . .
Si . . . . . . .
S ........
Ar . . . . . . .
Ca . . . . . .
Cr . . . . . . .
Fe . . . . . . .
Ni . . . . . . .
log
10
Abundance
8.09
6.78
8.03
6.92
8.00
7.21
6.56
6.81
6.12
8.12
6.70
NOTE.ÈCoronal abundances,
adapted from photospheric
abundances from Grevesse &
Anders (1991). The values are
those listed by Grevesse &
Anders, with 0.45 added for elements with ionization potentials
below 9 eV. The numbers given
are on the scale where hydrogen
has a logarithmic abundance of
12.
1012
JUDGE
Vol. 500
TABLE 2
TABLE 2ÈContinued
POTENTIALLY INTERESTING LINE LIST
Sequence
Transition
Be . . . . . . .
3Po ] 3Po
2
1
B ........
2Po ] 2Po
3@2
1@2
C ........
3P ] 3P
1
0
3P ] 3P
2
1
N ........
2Do ] 2Do
5@2
3@2
O ........
2Po ] 2Po
3P 3@2
] 3P 1@2
0
1
3P ] 3P
1
2
F ........
2Po ] 2Po
1@2
3@2
Mg . . . . . .
3P ] 3P
2
1
Al . . . . . . . .
2Po ] 2Po
3@2
1@2
Si . . . . . . . .
3P ] 3P
1
0
3P ] 3P
2
1
P ........
2Do ] 2Do
5@2
3@2
2Po ] 2Po
3@2
1@2
S .........
3P ] 3P
1
2
Ion
Na VIII
Mg IX
Al X
Si XI
S XIII
Ne VI
Na VII
Mg VIII
Al IX
Si X
S XII
Ar XIV
Mg VII
Al VIII
Si IX
S XI
Ar XIII
Ca XV
Mg VII
Al VIII
Si IX
S XI
Ar XIII
Ca XV
SX
Ar XII
Ca XIV
Ca XIV
Mg V
Al VI
Si VII
S IX
Ar XI
Ca XIII
Mg V
Al VI
Si VII
S IX
Ar XI
Ca XIII
Al V
Si VI
S VIII
Ar X
Ca XII
SV
Ar VII
Ca IX
Cr XIII
Fe XV
Ar VI
Ca VIII
Cr XII
Fe XIV
Ca VII
Cr XI
Fe XIII
Ni XV
Ca VII
Cr XI
Fe XIII
Ni XV
Cr X
Fe XII
Ni XIV
Ca VI
Cr X
Fe XII
Ni XIV
Ca V
Cr IX
Fe XI
j
(km)
6.225
4.064
2.753
1.904
1.030
7.631
4.674
3.028
2.044
1.430
0.7611
0.4411
9.031
5.846
3.928
1.920
1.018
0.5694
5.502
3.689
2.584
1.392
0.8360
0.5444
8.674
3.004
1.307
0.9122
5.607
3.659
2.481
1.252
0.6929
0.4087
13.53
9.113
6.513
3.754
2.617
2.258
2.905
1.964
0.9913
0.5534
0.3328
13.13
5.951
3.088
1.088
0.7347
4.544
2.321
0.8153
0.5303
6.153
1.806
1.0747
0.6702
4.086
1.551
1.0798
0.8024
4.260
2.217
1.282
17.98
3.103
1.561
0.8691
4.157
1.278
0.7892
Relative m(T ) É n /n
EL H
(Normalized to Fe XIV)
0.004
0.204
0.029
0.550
0.112
0.027
0.003
0.076
0.016
0.347
0.089
0.028
0.047
0.006
0.191
0.089
0.025
0.049
0.047
0.006
0.191
0.089
0.025
0.049
0.056
0.020
0.049
0.049
0.018
0.002
0.044
0.031
0.020
0.045
0.018
0.002
0.044
0.031
0.020
0.045
0.001
0.022
0.012
0.036
0.036
0.003
0.002
0.012
0.009
1.000
0.001
0.005
0.007
1.000
0.003
0.007
0.912
0.038
0.003
0.007
0.912
0.038
0.005
0.724
0.038
0.001
0.005
0.724
0.038
0.001
0.003
0.724
Sequence
Transition
3P ] 3P
0
1
Cl . . . . . . . .
2Po ] 2Po
1@2
3@2
Ar . . . . . . .
3P ] 3P
2
1
3F ] 3F
3
4
Ion
j
(km)
Relative m(T ) É n /n
EL H
(Normalized to Fe XIV)
Ni XIII
Ca V
Cr IX
Fe XI
Ni XIII
Cr VIII
Fe X
Ni XII
Fe IX
Ni XI
Fe IX
Ni XI
Fe IX
0.5116
11.48
5.786
6.098
19.29
1.010
0.6374
0.4231
1.867
1.279
2.855
2.571
2.218
0.035
0.001
0.003
0.724
0.035
0.001
0.457
0.027
0.457
0.027
0.457
0.027
0.457
3F ] 3F
2
3
NOTE.ÈThe last column lists the order of magnitude estimate of the
leading terms determining the line intensity, the product of the abundance
of the element and the relative di†erential emission measure at a temperature T \ 104(z ] 1)2 K, where z is the charge on the ion. These quantities are normalized such that lines of Fe XIV have the value unity (see
text). Wavelengths, which are from the NIST on-line database, are generally accurate to one part in 103 (see the Appendix).
ratios, high probabilities for photon collisional destruction,
and/or abundances, and they can be neglected. Two exceptions ([Fe XI] and [Ni XIII]) were included in the list because
of an interesting anomaly leading to very long wavelengths
for a particular transition (see the Appendix).
3.3. Intensity Calculations
3.3.1. Basic Equations
Ignoring stimulated emission, background sources of
emission, and other sources of emission along the LOS, the
frequency-integrated intensity of an optically thin line centered at rest wavelength j is given by
k
hc 1 0
n (s)A ds ergs cm~2 s~1 sr~1 , (1)
I \
j(k)
k
k j 4n
~=
k
where the symbols h and c have their usual meanings, A is
the Einstein A-coefficient of the line labeled k, and n (s)k is
j(k) k at
the population density of the upper level j(k) of line
position s along the LOS. The integration extends along the
LOS. Assuming that the plasma does not evolve rapidly
with time, the population densities are determined from rate
equations of the form [the label (k) is dropped henceforth]
P
; n (s)P (s) [ n (s) ; P (s) \ 0
j
ji
i
ij
jEi
jEi
and the particle conservation equation
(2)
n (s)
; n (s) \ el n (s) ,
(3)
i
n (s) H
i
H
where n (s) is the number density of hydrogen nuclei and
H is the abundance of the element relative to hydron (s)/n (s)
el
H
gen. In equation (2), P (s) is the total (collisional plus
ji
radiative) transition probability,
in units of s~1, at which an
ion in level j makes a transition to level i.
For the lines considered in Table 2, collisions between the
ions of interest and electrons and protons only need be
considered. Photospheric radiation must also be included
because it irradiates the corona, and the excitation prob-
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
abilities can be comparable to or larger than collisional
probabilities for certain optical and infrared transitions.
3.3.2. Ray Integrations and Temperature Distributions
Given a thermal model of the corona specifying electron
and proton densities and temperatures as a function of
radius (spherical symmetry is assumed here) and element
abundances and given the photospheric radiation Ðeld and
transition probabilities, it is straightforward to solve
numerically equations (2) and (3) and derive intensities
through equation (1). Such calculations su†er, however,
from a serious shortcoming. The LOS in the real Sun intersects plasmas at a wider variety of di†erent temperatures
than can be accounted for in any reasonable model
assuming spherical symmetry. Model calculations, in which
temperature is speciÐed only as a function of radius, cannot
reproduce adequately the distribution of material as a function of temperature along the computed LOS. These calculations would systematically skew results toward a few ions
that are preferentially formed at the chosen temperatures.
Therefore, a modiÐed approach is needed.
Consider the factorization of the population density n
j
into two components :
n
n \ n@ É ion ,
j
j n
el
(4)
where
1013
X (T , n, r). Equation (7) can then be split as follows :
k e
hc 1
0
I \
A
X (T , n, r)ds SY T ,
(8)
k j 4n k
k e
k
k
~=
where the quantity SY T must satisfy
k
0
0
X (T , n, r)Y (T )ds \
X (T , n, r)ds SY T , (9)
k e
k e
k e
k
~=
~=
i.e., SY T is deÐned to be the average value of the ionization
k
fraction to which the line k belongs, weighted by X (T , n, r),
k e
along the LOS. This is manifestly di†erent for all lines. If a
suitable approximation for SY T can be derived, then the
k
intensity for each line k can be calculated
using equation (8)
and not equation (7). Such an approximation can be estimated using the di†erential emission measure (e.g., Craig &
Brown 1976). For all lines of interest, the detailed calculations discussed below show that X (T , n, r) P na, where a
k can
e be written
is in the range 1 to 2. Then equation (9)
P
P
P
P
0
naY (T )ds \
k e
P
0
na ds SY T .
(10)
k
~=
~=
Transforming from an integration over path length, ds, to
one over logarithmic temperature T \ log T , one can
10 ethat
deÐne the di†erential emission measure m(T ) such
n2ds \ m(T )dT ;
(11)
then equations (9) and (11) give
n n n
n@ \ j el H n ,
(5)
j n n n
ion H
n is the population density of the ion to which line i
ion
belongs,
n is the number density of all ions of that element,
el
n is the number
density of hydrogen nuclei, and n is the
H
electron density. Then, including the label k again, equation
(1) becomes
P
0
n
hc 1
n@ (s) ion ds .
(6)
A
I \
j(k) n
k j 4n k
~=
el
k
The ionization fraction, n /n , implicitly including the
el electron temperature T
label k, is a strong functionionof the
and a weak function of all other parameters. It depends one
bound-free rate coefficients for processes that involve
mostly electron collisions and can be written Y (T ). In contrast, for all lines considered here, the term n@ k (s)e is a relaj(k)
tively weak function of T but is also a function
of the
e
incident radiation intensity, and thus the radius r, and on n.
It depends primarily on bound-bound rate coefficients and
thus can be written X (T , n, r). (The dependence of varie
ables T , r, and n on s is kimplicitly
assumed here). Then,
e
0
hc 1
X (T , n, r)Y (T )ds .
(7)
I \
A
k e
k e
k j 4n k
~=
k
As noted above, the term Y (T ) depends more sensitively
e
on T than the term X (T ).k Ionization
rate coefficients
e
k
e
determining Y (T ) scale as exp ([I/kT ), where I/kT ? 1.
e
e exp
Excitation ratek coefficients
determininge X (T ) scale as
k
e
([E/kT ), where E/kT > 1 for transitions between levels
with thee same principale quantum number. Thus, a distribution of material of various temperatures along the LOS
inÑuences the term Y (T ) with little inÑuence on the term
k e
P
P
P
m(T )na~2Y (T )dT \
m(T )na~2 dT SY T . (12)
k
k
*T
*T
Analysis of EUV emission lines from disk spectra of the
quiet Sun show that m(T ) has a peak near T \ 6.2, dropping
to 1/e of the peak intensity at T \ 5.9 and 6.5 (e.g.,
Raymond & Doyle 1981 ; Dere & Mason 1993). A reasonable approximation, given that disk spectra are heavily
weighted toward the coronal base anyway and that, therefore, there is little information from such spectra on the
form of m(T ) beyond roughly 1.2 R , is to assume that this
_ value, i.e., m(T ) ^ m ,
can be approximated by a constant
c
between T \ 5.9 and 6.5, and zero otherwise. While this
approximation ignores details of the peak in m(T ) near
T \ 6.2, it does yield a form for the expression for the line
intensity, which separates into the two simple factors in
equation (8). Using m(T ) \ m and *T \ 0.6 and ignoring
any weak dependence of na~2 con T , equation (12) yields
P
6.5
Y (T )dT .
(13)
k
5.9
Physically, SY T is the average value of the ionization fraction over the klogarithm of the electron temperature range
5.9È6.5. Ionization balance calculations solving for Y (T )
k
were made using rate coefficients of Arnaud & Raymond
(1992), modiÐed for Ðnite electron densities as discussed by
Judge et al. (1995), and values of SY T were determined for
each ion. They are tabulated in Tablek 3.
Line intensities were calculated as follows : values of SY T
k
were computed from an ionization balance calculation
ignoring all levels but those belonging to ground terms.
Then, a radial distribution of coronal density with radius
was adopted. The following formula was used as a simple
analytic Ðt to the radius r (in solar units, R \ 7 ] 1010
_
1
SY T ^
k
0.6
1014
JUDGE
Vol. 500
TABLE 3
COMPUTED INTENSITIES AT 1.1 R
_
Sequence
Transition
( j ] i)
Ion
j
(km)
Si XI
Si X
Mg VIII
Mg VII
A
(s~1)
SY T
k
0.071
0.142
0.073
0.030
n A
j ji
10 n n
ion
(cm3 s~1)
[log
n0A
j ji
10 n n
ion
(cm3 s~1)
[log
[
I
10
[1.43
0.73
[0.36
[2.34
[1.56
[0.17
[0.06
[0.04
[0.44
[0.19
[1.26
[0.30
[1.46
[3.31
[0.71
[2.42
[0.07
[2.02
[0.41
[2.62
[2.55
[0.97
[0.54
[3.85
[0.37
[2.42
1.36
1.35
[2.17
0.72
[0.77
[5.18
0.96
[1.68
[2.00
[1.79
1.12
[1.56
[1.24
[1.40
[2.03
[1.74
[2.03
[1.23
[0.56
[0.88
[1.72
[2.36
log
A B
d ln I ~1
dr
(R )
_
0.067
0.088
0.106
0.151
0.118
0.123
0.086
0.101
0.073
0.095
0.068
0.090
0.065
0.075
0.094
0.090
0.091
0.073
0.091
0.068
0.067
0.076
0.091
0.065
0.065
0.096
0.073
0.091
0.068
0.068
0.095
0.109
0.084
0.103
0.065
0.064
0.074
0.065
0.065
0.065
0.065
0.065
0.066
0.091
0.091
0.070
0.065
0.065
3Po
1.933
1.80(0)
9.84
9.80
2?1
2Po
1.430
2.97(0)
8.19
8.41
3@2?1@2
2Po
3.03
3.10([1)
8.74
8.76
3@2?1@2
C ........
3P
9.03
2.55([2)
9.96
9.95
1?0
3P
5.50
8.24([2)
9.33
9.33
2?1
Si IX
3P
3.93
0.169
3.03([1)
8.81
8.84
1?0
3P
2.585
7.91([1)
8.67
8.82
2?1
S XI
3P
1.920
0.154
2.51(0)
8.10
8.55
1?0
3P
1.393
5.08(0)
8.56
8.88
2?1
Ar XIII
3P
1.016
0.121
1.72(1)
7.77
8.67
1?0
3P
0.8300
2.25(1)
8.83
9.06
2?1
Ca XV
3P
0.5695
0.033
9.56(1)
7.79
8.52
1?0
3P
0.5445
8.28(1)
8.89
8.91
2?1
N ........
SX
2Do
8.68
0.225
2.42([2)
10.81
10.79
5@2?3@2
O ........
Si VII
3P
2.481
0.027
1.52(0)
8.68
8.94
1?2
3P
6.51
2.02([1)
9.96
9.96
0?1
S IX
3P
1.252
0.178
1.14(1)
8.36
8.98
1?2
3P
3.76
9.81([1)
9.77
9.75
0?1
Ar XI
3P
0.6918
0.184
6.74(1)
8.32
9.13
1?2
3P
2.619
2.84(0)
9.88
9.84
0?1
Ca XIII
3P
2.258
0.155
4.16(0)
10.04
9.99
0?1
F ........
S VIII
2Po
0.9916
0.087
1.86(0)
9.00
9.11
1@2?3@2
Ar X
2Po
0.5522
0.208
1.06(2)
8.60
9.46
1@2?3@2
Mg . . . . . .
Fe XV
3Po
0.5023
0.098
7.80([3)
13.10
13.09
2?0
3Po
0.7062
3.37(1)
9.46
9.46
2?1
Al . . . . . . . .
Ca VIII
2Po
2.321
0.006
7.19([1)
8.58
8.77
3@2?1@2
Fe XIV
2Po
0.5304
0.087
6.02(1)
7.84
7.97
3@2?1@2
Si . . . . . . . .
Fe XIII
3P
1.075
0.111
1.40(1)
7.71
8.11
1?0
3P
0.5386
6.35([3)
11.45
11.50
2?0
3P
1.080
9.87(0)
8.25
8.31
2?1
S .........
Ni XIII
3P
0.5117
0.073
1.57(2)
8.56
9.78
1?2
3P
19.30
6.84([3)
11.43
11.42
0?1
Fe XI
3P
0.7894
0.100
4.38(1)
8.16
8.49
1?2
3P
6.08
2.26([1)
9.98
9.95
0?1
3Fo ] 3Do
0.6936
5.81(0)
11.12
11.12
4
3
3Go ] 3Fo
0.5203
6.31(0)
11.02
11.03
4
4
Cl . . . . . . . .
Fe X
2Po
0.6376
0.108
6.94(1)
8.10
8.26
1@2?3@2
4F
1.945
4.26(0)
10.26
10.28
7@2?9@2
2F ] 4F
0.5542
6.12(0)
10.48
10.48
7@2
7@2
2F ] 4F
0.7076
5.43(0)
10.53
10.53
7@2
5@2
2G ] 2F
1.009
4.69(0)
11.01
11.01
9@2
7@2
2G ] 2F
0.9762
6.10(0)
10.74
10.74
7@2
7@2
Ar . . . . . . .
Fe IX
3Po
1.868
0.110
2.07(0)
10.77
10.76
2?1
3Fo ] 3Po
0.6393
5.06([2)
10.52
10.70
3
2
3Fo
2.856
1.07(0)
9.19
9.37
3?4
3Fo
2.218
3.11(0)
9.55
9.78
2?3
1Fo ] 3Do
0.9788
7.77(0)
10.73
10.73
3
3
1Fo ] 1Do
1.102
2.01(0)
11.32
11.32
3
2
NOTE.ÈIntensities are in units of ergs cm~2 s~1 st~1. Einstein A-coefficients use the notation x É y(z) 4 x É y ] 10z. Wavelengths, from the
CHIANTI database, are generally accurate to one part in 103 (see the Appendix).
Be . . . . . . .
B ........
cm), from density data tabulated by Allen (1973) ° 84 :
C
A
n(r) \ n a É exp [
0
A
B
r[1
h
] 1[a[
B
A
B
D
1
r[1
1
É exp [
]
.
b
dÉh
br3
(14)
Parameters applicable to the solar minimum equatorial
coronal density data were adopted : n \ 2 ] 108 cm~3,
0
a \ 0.8, b \ 30, d \ 3. This yields densities
that match
AllenÏs tabulated values to within 0.1 dex and that lie
between the tabulated solar minimum polar and solar
maximum densities of Allen (1973). In this loose sense, the
densities used represent an average coronal density struc-
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
1015
ture. The scale height h corresponds to an atmosphere in
hydrostatic equilibrium at 106 K. The cubic term represents
a solar wind component accelerating as v(r) P r. Using these
densities, Ðxed abundances were adopted (Table 1), an electron temperature of 106 K was used, and level populations
for levels within each ion were computed to determine
X (T , n, r) as a function of radius, out to 20 R . LOS
k e
_
integrations were performed numerically to determine
/0 X (T , n, r) ds in equation (8), and the intensities were
~= k e
evaluated by this equation.
Note that the approximation embodied in equation (8) is
intended to be used for calculation of average intensities as
a function of projected height above the solar limb. Several
data sets show that this approximation must be invalid for
studying particular coronal structures (see Penn et al. 1994
and references therein).
The largest sources of error arise in the use and calculation of the term SY T. First, the separation of the integral
k
equation (7) into the form of equation (8) of course makes
the assumption that this is meaningful for all lines. Furthermore, there is the crucial assumption that the di†erential
emission measure is constant between logarithmic temperatures of 5.9 and 6.5, and zero outside this range, and
that this is valid at all radial distances in the corona. Thus,
even if the ionization balance as a function of temperature is
free of error (this is not true), these assumptions are unlikely
to be correct at least at all heights in the corona, and potentially large, systematic errors may arise. At this stage, it
seems best to simply recognize this source of uncertainty
and try not to overinterpret comparisons between certain
ions and their behavior as a function of radius.
3.3.3. Calculations of L evel Populations within Each Ion
The transitions in Table 2 belong to sequences isoelectronic with the elements Be, B, C, N, O, F, Mg, Al, Si, P, S,
Cl, and Ar. The Ðrst detailed calculations were therefore
done for the ions of each isoelectronic sequence that are, on
the basis of Table 2, expected to be brightest. In order of
isoelectronic sequence (equivalently number of electrons
around the nuclei), these are Si XI, Si X, Si IX, S X, S IX, Ar X,
and Fe XV through Fe IX. The resulting line intensities and
their gradients with projected height above the limb at
r \ 1.1 R are listed in Table 3, together with additional
_
ions discussed
below. The table lists components of
equation (8) and includes intensities computed both with
and without the inÑuence of photospheric radiation in the
calculations of X (T , n, r).
e dependence of the intensities I of the
Figure 1 showsk the
k s~1
10 strongest lines (ordered using units of photons cm~2
st~1, since modern IR detectors count photons) as a function of projected radial height above the solar limb (upper
panel). In the lower panel, the ratio of the intensities computed with photospheric radiation to those computed
without are shown. Inspection of the results in the table, for
the ions representing the strongest transitions of speciÐc
isoelectronic sequences, reveals the following results :
Number densities n@ [equivalently X (T , n, r)] were
j
k eincludes levels
obtained for a given atomic
model, which
from just an individual ion (details below), for speciÐc electron densities, temperatures, and radiation Ðelds. The incident photospheric radiation was computed using a Planck
function at 5900 K, multiplied by the usual dilution factor
W\
C S A BD
1
1[
2
1[
R 2
_
,
r
(15)
where R is the solar radius. Care was taken to include
_
excited energy
levels in each ion that can inÑuence the IR
forbidden lines through electron collisional excitation followed by radiative decay. Thus, all levels with the same
principal quantum number as the ground level were
included for all ions, and all higher levels were ignored.
Calculations were made using the latest available rate coefÐcients. Extensive use was made of the CHIANTI compilation (Dere et al. 1997), which includes electron
collisional processes largely from the volume edited by
Lang (1994) and radiative transition probabilities from a
variety of sources. Forbidden radiative transition probabilities were taken from Kaufman & Sugar (1986), with the
exception of ions of the Be sequence, which were taken from
Brage et al. (1997). While proton collisions are known to be
signiÐcant for some ions (e.g., Flower & Pineau des Foreüts
1973 ; Sahal-Brechot 1974), their inÑuence for electron temperatures of 106 K and above is relatively small for IR
transitions compared with electron collisions and radiative
excitation, and they have been neglected in these calculations.
3.3.4. Uncertainties
Uncertainties in these calculations are difficult to assess
accurately because errors arise both in the atomic parameters a†ecting level populations and in our understanding of
the coronal plasma properties, especially the distribution of
emitting material with temperature. Aside from negligible
errors in atomic constants (including transition
wavelengths), the smallest errors almost certainly arise from
the term X (T , n, r). With some potential exceptions disk ethe maximum likely error in these terms is a
cussed below,
factor of 2, and more likely closer to 10%È20%, on the basis
of error estimates in collisional excitation rate coefficients
(e.g., see the discussions in the volume edited by Lang 1994)
and in forbidden line radiative transition probabilities
(Kaufman & Sugar 1986).
3.3.5. Results
1. As anticipated in the Appendix, forbidden lines arising
from transitions between excited levels are substantially
weaker than those arising from transitions between the
ground levels (compare the IR line of Si XI, a transition
between excited levels in the Be isoelectronic sequence, with
other ions of silicon in the table, for example). In particular,
all forbidden lines from ions of the Be, N, and P sequences
are more than a factor of 10 weaker than the strongest lines,
and those of the Mg and Ar sequences are a factor of 10 or
so weaker. These and other ions of these isoelectronic
sequences can therefore be discounted as candidates for
Zeeman measurements.
2. The excitation of essentially all remaining forbidden
lines, i.e., of the B, C, O, Al, Si, S, and Cl sequences, have
strong contributions from photoexcitation by the photospheric radiation. This is indicated by a comparison
between the calculations with and without this process (Fig.
1, lower panel).
3. The largest inÑuence of photoexcitation occurs in excitation of the lowest transition2 (J \ 1 ] 0) in ions of the C
and Si sequences and the lowest transition (J \ 1 ] 2) in
2 Here and henceforth, the notation used is that of higher level ] lower
level.
1016
JUDGE
Vol. 500
FIG. 1.ÈDependence of the computed line intensities for the 10 strongest lines (evaluated at r \ 1.1 R ; in units of photons s~1 st~1) in the visual and IR
_ the inclusion of photospheric radiation in the
regions of the solar coronal spectrum. L ower panel : ratio of the intensities computed with and without
excitation calculations (i.e., the P and P terms in eq. [2]). The radial behavior of the electron density n is also shown (thick line) in the upper panel. Note
ij are strongly
ji
that the intensities of some lines that
a†ected by photoexcitation, for instance that of Fe XIII at 1.0747 km, follow a radial dependence similar to n
to the Ðrst power, while others (e.g., Fe XIII 1.0798 km) behave more like n to the second power.
the O and S sequences. These transitions might therefore be
expected to give a strong Ñuorescent polarization signal
that is comparable to or stronger than the several tens of
percent computed for the equivalent (J \ 1 ] 0) transition
in Fe XIII (Si-like) by Sahal-Brechot (1977). Additional calculations would be needed to assess the expected degree of
linear polarization for each ion.
4. The magnitudes of the computed forbidden line intensities of these particular lines are in satisfactory agreement
with those measured by, e.g., Je†eries, Orrall, & Zirker
(1971) at optical wavelengths and by Kuhn, Penn, & Mann
(1996) at IR wavelengths. More careful comparisons with
observations are made below.
3.3.6. Additional Calculations and Spectral Synthesis
Encouraged by the initial success of these Ðrst results,
detailed calculations were performed for the other abundant ions of Table 2. Based upon the detailed results above,
only those ions belonging to the B, C, O, Al, Si, S, and Cl
sequences that appear in Table 2 are likely to be strong
enough to warrant further study. The results of these additional calculations are also listed in Table 3 and are presented graphically in the synthetic spectra shown in Figure 2.
The Ðgure includes contributions to the intensities from the
K corona at a level of 1/106 of the solar photospheric intensity, which is a reasonable estimate of typical conditions at
some of the best high-altitude coronagraphic sites, for a
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
1017
FIG. 2.ÈSynthetic emission line spectra evaluated at 1.1 and 1.3 R and including typical intensities of the K corona (the F corona is negligible at these
_ cooled to liquid nitrogen temperatures (a 77 K blackbody spectrum). The unit of
heights) from MuŽnch (1966) and thermal emission from an instrument
intensity is ergs cm~2 s~1 AŽ ~1 sr~1, and line intensities plotted are peak intensities assuming a line width (FWHM) of 30 km s~1 in Doppler units (thus, peak
intensities are approximately integrated intensities I divided by wavelength j measured in microns). No other source of IR light is included in this plot (e.g.,
k
k
from thermal emission/scattering from the EarthÏs atmosphere).
Line identiÐcations
are marked (arbitrarily) for lines whose peak intensities exceed 20% of
the neighboring background intensity.
wavelength near 1 km (e.g., Golub & Pasacho† 1997, pp.
143 and 147).
The list of ions in Table 3 should contain all of the strong
coronal emission lines between 1 and 10 km. Figure 2
should similarly contain all of the strongest coronal emission lines. Atomic data were from the sources discussed
above, except that electron excitation collisional data for Ca
VIII were taken from the homologous ion Mg VIII, since no
data for this ion could be found. For Ni XIII, collisional data
for Fe XI were adopted in the absence of any speciÐc data.3
Aside from illustrating the strongest coronal transitions
3 This could be substantially in error. Ions of the S sequence require
special attention since the J \ 1 and 0 levels cross over between Ni XIII and
Cu XIV, owing to the transition from LS to jj coupling. Furthermore, the Ni
XIII 0.512 AŽ line is computed to be an order of magnitude weaker than
observed (see ° 4). The data used here are based upon perturbation theory
in the SchroŽdinger formulation of quantum mechanics. More detailed relativistic atomic structure calculations appear to be warranted for these ions
(T. Brage 1995, private communication).
1018
JUDGE
TABLE 4
Vol. 500
4.
CANDIDATE LINES FOR THE APPLICATION OF
STOKES POLARIMETRY TO DETERMINE
CORONAL MAGNETIC FIELDS
Ion
Wavelength
(km)
Fe XIV . . . . . . . . . . . . . . . . . . . . . . . . .
Fe X . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fe XI . . . . . . . . . . . . . . . . . . . . . . . . . .
Fe XIII . . . . . . . . . . . . . . . . . . . . . . . . .
Fe XIII . . . . . . . . . . . . . . . . . . . . . . . . .
Si X . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S XI . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Si IX . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fe IX . . . . . . . . . . . . . . . . . . . . . . . . . .
Mg VIII . . . . . . . . . . . . . . . . . . . . . . . .
Si IX . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mg VII . . . . . . . . . . . . . . . . . . . . . . . .
Mg VII . . . . . . . . . . . . . . . . . . . . . . . .
Fe XI . . . . . . . . . . . . . . . . . . . . . . . . . .
Ni XIII . . . . . . . . . . . . . . . . . . . . . . . . .
0.530
0.637
0.789
1.0747
1.0798
1.430
1.920
2.584
2.855
3.028
3.929
5.502
9.031
6.081a
19.3a
a These lines are computed to be very weak ;
they are included because of their favorably
long wavelengths and potential problems with
the atomic data used (see text).
present in the IR spectrum, Figure 2 illustrates a fundamental trend. The strongest lines (measured either in photon or
energy units) lie in the optical and near-IR regions below
D1 km. Lines become substantially weaker at longer wavelengths (with the previously noted caveat that the interesting S sequence ions of Fe XI and Ni XIII require further
work). Those beyond 10 km (see Table 2 for many
examples) are probably too weak for the high signal-tonoise ratio requirements for magnetograph measurements.
The physical reasons for this are simple. Almost all of the
transitions discussed here are magnetic dipole (M1) transitions. Atomic line strengths being equal (they do not
depend strongly on ion charge for M1 transitions), the Einstein A-coefficients scale with transition wavelength j as
k
j~3. Transitions induced by electron collisions tend to have
k
higher probabilities for M1 transitions which lie further in
the IR region, simply because for a given isoelectronic
sequence, the collisional probabilities are largest for ions of
lowest charge, i.e., those having the longest wavelengths in
M1 transitions. Photon escape in other (electric dipole)
transitions sharing a common upper level, when present, is
more likely for longer wavelength transitions for similar
reasons. Thus, the probabilities of photon escape in a given
M1 line tend to decrease rapidly with increasing j . Furk with
thermore, radiative excitation probabilities decrease
increasing wavelength j . Einstein B-coefficients are independent of j for a Ðxed katomic line strength, but the intenk
sity of the incident
photospheric radiation varies according
to j~1 in the Rayleigh-Jeans limit. All of these processes
servek to produce the sharp decrease in the intensity of IR
coronal lines with wavelength seen in Figure 2.
Finally, on the basis of Figure 2, Table 4 lists the most
likely candidates for application to Stokes measurements of
the Zeeman e†ect, assuming that the calculations are consistent with available observations. Note that the wavelengths are generally not reliable to better than three
signiÐcant Ðgures (see the Appendix).
INITIAL COMPARISON WITH OBSERVATIONS
Detailed comparisons of observed and computed intensities are not yet warranted given that the adopted density
structure is just a spatial ““ average ÏÏ that probably cannot
exist at any given time on the Sun. Furthermore, no complete spectrum exists that simultaneously covers the optical
and IR region. Nevertheless, comparisons of computed
intensities with those that are available can be made to see if
the calculations are in rough agreement in terms of average
intensities and to see if their measured dependence with
projected height above the solar limb is reasonable.
There are several sources of optical and IR data. Between
0.3 and 0.8 km, Je†eries et al. (1971) list line intensities
measured at two heights in ““ coronal condensations ÏÏ seen
at the 1965 May 30 total eclipse. Kuhn et al. (1996) list
intensities (and useful upper limits) of lines between 1 and
1.5 km from the total eclipse of 1994 November 3 that may
be considered to be from a similar type of coronal condensation as that measured by Je†eries et al. (1971). The
observations of Kuhn et al. (1996) are probably of the
highest quality and should be given extra weight. Finally,
beyond 1.5 km, the only published work to claim detections
are the 1È3 km grating data of MuŽnch et al. (1966), from the
1996 November 12 total eclipse, and FTS data from Olsen
et al. (1971), from the 1970 March 7 total eclipse. Note that
the intensities of the latter are questioned by Penn & Kuhn
(1994), who suggested that these early measurements overestimated line intensities.
Figure 3 shows a comparison between observed and
computed data, taking all data at face value. For each set of
observations, the computed intensities were evaluated
through lines of sight passing through the radii in the POS
marked in the Ðgure (so a value of 1.03 R was used for the
_
data of MuŽnch, Neugebauer, & McCammon
1966, for
example). Considering the levels of approximation in the
calculations (see ° 3.3.4) and the diverse set of observations,
the observed and computed data can be regarded as being
consistent with one another (note that the Ðgure is plotted
in the log-log plane). A scatter of a factor of 2 is seen around
equality, with the exception of [Ni XIII] at 0.512 km, which
is an order of magnitude too weak in the calculations compared with iron lines (data are from Je†eries et al. 1971).
This may result from inadequate treatment of collision
strengths and transition probabilities for ions in the S
sequence (discussed above). [S XI] 1.392 km is computed to
be an order of magnitude stronger that might be inferred
from a glance at Table 1 of Kuhn et al. (1996). It is not
shown in the Ðgure since the data tabulated by Kuhn et al.
(1996) are upper limits to intensities without correction for
atmospheric absorption, and at this wavelength it is nearly
100% (J. Kuhn 1997, private communication).
One more important comparison of the calculations with
observations can be made. Kuhn et al. (1996) show, in their
Figure 3a, the dependence of the intensity of the [Fe XIII]
1.0747 km line with projected height above the lunar limb.
This can be directly compared with the calculations shown
in Figure 1. Close to the solar limb, the 1/e scale height of
the observed intensity with projected height is D0.18 R .
In the calculations, this is D0.09 R . The intensity _
of
_
[Fe XIII] lines is computed to vary roughly
with electron
density na, where a is 1 or a little higher (see Fig. 1). This
discrepancy suggests several possibilities. It is likely that the
density structure at the time of the observations di†ers sig-
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
1019
FIG. 3.ÈScatter plots showing observed and computed frequency-integrated intensities of IR lines. Data are from a heterogeneous set of observations,
and computations are from an ““ average ÏÏ coronal density distribution, so comparisons can be made only roughly. Error bars are difficult to assess in both
observed and computed data (see text).
niÐcantly from the ““ average ÏÏ structure represented by
equation (14). There may be a problem with the interpretation that the radial fallo† of the Fe XIII line intensity
observed by Kuhn et al. (1996) has a completely solar
origin. The latter probably cannot be completely discounted, given the concerns expressed by Kuhn et al. (1996)
about the presence of cirrus clouds during the observations.
Another more serious possibility is that something crucial is
missing from the calculations. The most likely candidate is
the assumed dependence of the di†erential emission
measure on projected height (see ° 3.3.4).
In conclusion, the agreement between observed and computed intensities at a speciÐc height in the corona is satisfactory. Where comparisons are possible, relative intensities
between di†erent lines are in quite good agreement with just
one noted exception. It is reasonable to assume that the
theoretical calculations can be used for their primary task
speciÐed in ° 1 : to identify those strong lines that should be
considered as candidates for Zeeman determinations of the
coronal magnetic Ðeld.
The lack of agreement of computed and observed radial
dependence for the Fe XIII 1.0747 km line is more worrisome, but the current uncertainties in treatment of emission
measures and in the interpretation of data acquired under
variable atmospheric absorption conditions preclude a
deÐnitive conclusion.
5.
SUMMARY AND CONCLUSIONS
New calculations of the intensities of the forbidden line
1020
JUDGE
spectrum of the corona have been presented using the best
currently available atomic data. Several transitions of the B,
C, O, Si, and S isoelectronic sequences are sufficiently
strong, at least at wavelengths up to 4 km, that they are
potentially of great interest as diagnostics of the coronal
magnetic Ðeld through the Zeeman e†ect. Most of these
lines also have the desirable property of falling relatively
slowly with projected height above the solar limb (see Fig.
1). They are listed in Table 4. Computed intensities are in
reasonable agreement with the observed coronal spectrum,
where such data are available. Ions of the S sequence, especially Fe XI and Ni XIII, deserve new atomic structure calculations in the relativistic formalism.
Half of the most promising lines listed in Table 4 have not
been observed before and warrant observational study.
Most of these lie in atmospheric transmission windows and
could in principle be observed from the ground. Portions of
the coronal spectrum between 1 and 10 km will be acquired
by a joint NSO/NCAR/Rhodes College/Lindau experiment
during the 1998 February 26 total eclipse to search for some
Vol. 500
of the strongest predicted lines. An important aim will be to
determine the radial dependence of the IR emission of at
least one IR line, simultaneously with a determination of
the radial dependence of the electron density from the polarized brightness of white light. Such measurements are
needed to assess the completeness and accuracy of the theoretical intensities computed here. Similar work is worthwhile using existing eclipse data.
Future work should also include attempts to determine
the emission measure distribution as a function of projected
height above the limb for a variety of structures. Existing
data from the CDS and SUMER instruments on SOHO
would certainly allow such determinations, at least for low
projected heights above the solar limb.
The author acknowledges the assistance of R. Meisner for
assistance with developing atomic models and thanks Je†
Kuhn and Matt Penn for useful discussions. B. C. Low and
Roberto Casini provided useful comments on the manuscript.
APPENDIX A
FINDING STRONG FORBIDDEN LINES IN THE INFRARED SPECTRUM OF THE CORONA
In this Appendix, a Ðrst cut through the search for suitable lines is described. Elimination of candidate lines was done on the
basis of ion charge, abundance, types of transitions, and considerations of atomic processes that determine the strength of the
transitions. More than 50 potentially interesting lines with j [ 1 km survive (Table 2) and are discussed in more detail in
the main text.
A1.
ABUNDANCE CONSIDERATIONS
Given the measurements of the [Fe XIII] line reported by Kuhn (1995) and that this is one of the strongest forbidden lines in
the coronal spectrum, the search can be limited to lines of elements whose ““ coronal ÏÏ abundances (e.g., Grevesse & Anders
1991) are at least 1/100 of that of iron, i.e., H, He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Cr, Fe, and Ni. The Ðrst Ðve elements
are not useful because they contain no suitably strong IR emission lines (see below). Adopted abundances for the remaining
elements are listed in Table 1.
A2.
ELIMINATION OF TYPES OF TRANSITIONS
IR transitions in highly ionized ions occur between levels of the same terms (forbidden transitions) or between highly
excited Rydberg states (allowed transitions). The latter will have extremely weak intensities, since the levels will essentially be
populated only by dielectronic recombination, which has rate coefficients several orders of magnitude smaller than the rate
coefficients for collisional excitation. Thus, only the forbidden transitions in ground terms and metastable terms need further
consideration, since other levels will have negligible populations due to the rapid decay to the ground levels in permitted
transitions. This is corroborated by the exploratory work of Chang & Deming (1995).
There are thus just two types of forbidden transitions of interest here. The Ðrst are transitions between levels in the ground
term of the ions. These belong to the B, C, O, and F and the homologous Al, Si, S, and Cl sequences and include the
well-studied [Fe XIV] 0.5304 km green line (Al sequence) and the [Fe XIII] 1.0747 km line (Si sequence). The second are
transitions between levels of excited metastable terms and include transitions of the He, Be, Ne, Ar, and Mg sequences (triplet
metastable terms with opposite parity to the singlet ground term), and N and P sequences (doublet metastable terms of the
same parity as the quartet ground term). Transitions of the Ðrst kind were treated fully (for wavelengths longer than 1 km) by
Greenhouse et al. (1993), but the second were treated less completely.
Forbidden transitions of the second kind belonging to ions of the He, F, and Ne isoelectronic sequences are expected to be
much weaker than transitions belonging to the Ðrst kind. This is because the metastable levels belong to an electron
conÐguration with principal quantum number one greater than that of the ground conÐguration. Such levels lie at energies
greatly in excess of the mean thermal energy under coronal ionization equilibrium conditions.4 (The same is not true for ions
4 In coronal ionization equilibrium, the thermal energy where an ion reaches maximum abundance scales approximately as kT P I , where I is the
z º 20. Energies
z
ionization potential of an ion of charge z [ 1 (z \ 1 for neutrals, z \ 2 for singly ionized ions, etc.), and I P z2 and is in a regime wheree I /kT
z momentum and spin quantum
z numbers,
e
E of metastable terms with the same principal quantum number n as the ground term, but di†erent angular
scale as
n residual electrostatic term in the Hamiltonian, which scales as z1, and are thus formed in a regime where E /kT D 10/z. Energies E
the
of metastable terms
n e
n`1
with principal quantum number n ] 1 scale as z2 and are thus formed in a regime where E /kT D 10.
n`1 e
No. 2, 1998
LINES FOR POLARIZATION MEASUREMENTS. I.
1021
FIG. 4.ÈDependence of the energy level splittings of Ðne structure levels, plotted as the transition wavelength in km, as a function of ion charge z along
the sulfur isoelectronic sequence. (The energy di†erences in ergs are simply 104hc/j). The behavior of the plus symbols shows ““ normal ÏÏ isoelectronic
behavior. The behavior of the asterisks does not. In this case, the J \ 0 and 1 levels actually cross between Ni XIII and Cu XIV, yielding anomalously long
wavelengths for transitions between the J \ 0 and 1 levels for many ions in this sequence.
of the homologous Cl and Ar sequences, which have metastable quartet levels that have the same principal quantum number
as the ground levels).
Transitions of the Ðrst kind will in general be stronger than transitions of the second kind for other reasons. The strength of
forbidden transitions of the second kind depends primarily on whether the metastable levels have the same parity as the
ground term. If not, the upper metastable levels can also decay via intercombination (IC) electric dipole (E1) transitions. For
such cases, such as the nsnp2 4P J \ 1/2, 3/2, 5/2 metastable levels in the B and Al sequences (the levels all have IC transitions
J ground term, ns2np2Po
to levels of the opposite parity
), the M2 forbidden lines (e.g., nsnp2 4P ] nsnp2 4P ) are
1@2,3@2
5@2 database 3@2
completely negligible. For example, data in the National
Institute of Standards and Technology (NIST)
for the
Al-like ions Fe XIV and Ni XVI show that the M1 transition probabilities between levels of the 3s3p2 4P term are 6 or more
orders of magnitude smaller than the competing electric dipole (E1) intercombination transition probabilities. Given the
stronger dependence of the M1 transition probabilities on ion charge z than the intercombination E1 transition probabilities
(e.g., the Be- and Mg-like ions are discussed by Brage et al. 1998), these transitions can be safely ignored for all ions of less or
equal charge considered here. This lends support to the omission of such terms in the study of Greenhouse et al. (1993).
Metastable levels that have the same parity as the ground levels, or which have opposite parity but cannot decay via IC
transitions owing to the J selection rules, decay via M1 and higher order multipole transitions. The IR transitions between
levels of such terms are thus likely to have much higher branching ratios. An example is in the Be isoelectronic sequence where
the radiative decay routes include just the magnetic quadrupole (M2) transition 2s2p3Po ] 2s2 1S of the 2s2p 3P level is
2 8.5 Table
0 3 of Greenhouse
2 et al.
mostly via the forbidden IR transition to the 2s2p 3P level for nuclear charge greater than
1
(1993) lists branching ratios for this and similar transitions, but they omitted the Ar isoelectronic sequence.
One last aspect of the calculations of Greenhouse et al. (1993) is very useful for reducing the list of lines further, at least for
transitions of the Ðrst kind. In their Figure 3 they present ““ critical electron densities,ÏÏ n , i.e., those electron densities at
which collisional deexcitation rates of the upper level of a line equal radiative decay ratescrit(their eq. [1]). The line emission
coefficient scales with the electron density of the emitting plasma according to n /(n ] n ) É n2. The solar corona typically
e
has electron densities n \ n D 108 cm~3. Thus line intensities with n > n crit
scalecrit
as n en , which
is >n2 .
e
cor
crit
cor
crit cor
cor
A3.
LIST OF PROMISING LINES
Based upon the above criteria, the list of potentially interesting lines is given in Table 2. In summary, it basically consists of
lines of ions of all abundant elements (Ne, Na, Mg, Al, Si, S, Ar, Ca, Cr, Fe, and Ni) that have charges between 4 and 14, that
belong to transitions between ground or metastable levels of the Be, B, C, N, O, F, Mg, Al, Si, P, S, Cl, and Ar isoelectronic
sequences, and that also have large branching ratios and small collisional destruction probabilities.
To derive this a compilation of the energies of atomic levels for all abundant elements was done using the NIST on-line
database.6 Plots of energy separation between the relevant levels as a function of nuclear charge Z along each isoelectronic
5 The Einstein A-coefficients of the forbidden transitions scale with Z to a higher power than the spin-changing transitions owing to the dependence of the
transition matrix elements and wavelengths on Z (e.g., Cowan 1981).
6 The NIST on-line database, which contains nonrefereed material and is not maintained by the Astrophysical Journal, can be found at
http ://aeldata.phy.nist.gov/nist–atomic–spectra.html.
1022
JUDGE
sequence were made, and we checked for obvious departures from smooth behaviors (none were found). Some missing data
were interpolated using such plots. One interesting isoelectronic plot, for the S sequence, is shown in Figure 4.
The ““ normal ÏÏ behavior, qualitatively exhibited by the J \ 2 ] 1 transition in Figure 4, shows that the wavelength of a
forbidden line decreases smoothly and monotonically with nuclear charge A along an isoelectronic sequence. This follows
from the fact that the relativistic terms in the atomic Hamiltonian gradually increase along a sequence. In the isosequences
studied here, systematic departures were found for the N and the S sequences, which have wavelengths that are not monotonic
with A. For the S sequence, this occurs because the energy levels in fact cross over (i.e., switch order), and near the crossing
point, the energy splittings are smaller than they should be. This explains why the 3p4 3P ] 3p4 3P transitions in Fe XI and
0
1 simple extrapolation.
Ni XIII have wavelengths closer to 10 and 20 km than to 1 or 2 km, as might have been expected
from
These abundant ions therefore fortuitously have much longer wavelengths than they should, owing to a quirk in the atomic
physics of their isoelectronic sequence. These will therefore potentially be of interest for Zeeman measurements if they prove
to be sufficiently intense.7 Calculations for these ions were therefore treated with special care (see the main text).
It should be noted that the wavelengths tabulated, which were taken from the NIST database, are typically accurate only to
one part in 103 for lines between 1 and 20 km (see e.g., Greenhouse et al. 1993). Much more accurate data are becoming
available through new measurements of low-density astrophysical objects (e.g., Oliva et al. 1994 ; Feuchtgruber et al. 1997),
yielding accuracies of one part in 104 or better.
7 Note that Ferland (1993) found this same physical e†ect to be responsible for causing population inversions in the J \ 0 ] 1 transitions in the Fe XI ion
in calculations of photoionized plasmas.
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